Definition:Min Operation

Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

The min operation is the binary operation on $\struct {S, \preceq}$ defined as:

$\forall x, y \in S: \map \min {x, y} = \begin {cases} x & : x \preceq y \\ y & : y \preceq x \end {cases}$

General Definition

Let $S^n$ be the cartesian $n$th power of $S$.

The min operation is the $n$-ary operation on $\struct {S, \preceq}$ defined recursively as:

$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in S^n: \map \min x = \begin{cases} x_1 & : n = 1 \\ \map \min {x_1, x_2} & : n = 2 \\ \map \min {\map \min {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\ \end{cases}$

where $\map \min {x, y}$ is the binary min operation on $S^2$.

Notation

The notation $\min \set {x, y}$ is frequently seen for $\map \min {x, y}$.

This emphasises that the operands of the min operation are undifferentiated as to order.

Some sources use the notation $x \wedge y$ for $\map \min {x, y}$.

Also see

• Definition:Max Operation

• Results about the min operation can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.6$